# Curve-Curve Intersection¶

The problem of intersecting two curves is a difficult one in computational geometry. The Curve.intersect() method (when using the GEOMETRIC strategy) uses a combination of curve subdivision, bounding box intersection, and curve approximation (by lines) to find intersections.

## Curve-Line Intersection¶

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.0, 0.375],
...     [1.0, 0.375],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=1)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.25, 0.25],
[0.75, 0.75]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0.25 , 0.375],
[0.75 , 0.375]])

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.5, 0.0 ],
...     [0.5, 0.75],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=1)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.5 , 0.6666...]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0.5, 0.5]])

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [4.5, 9.0],
...     [9.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.0, 8.0],
...     [6.0, 0.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=1)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.3333..., 0.5 ]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[3., 4.]])

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.375],
...     [1.0, 0.375],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=1)
>>> nodes2 = np.asfortranarray([
...     [0.5, 0.0 ],
...     [0.5, 0.75],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=1)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.5, 0.5]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0.5 , 0.375]])

>>> nodes1 = np.asfortranarray([
...     [-1.0, 1.0],
...     [ 0.5, 0.5],
...     [ 0.0, 2.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [ 0.5 , 0.5 ],
...     [-0.25, 1.25],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=1)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.5 , 0.6666...]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0., 1.]])


## Curved Intersections¶

For curves which intersect at exact floating point numbers, we can typically compute the intersection with zero error:

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.0,  0.75],
...     [0.5, -0.25],
...     [1.0,  0.75],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.25, 0.25],
[0.75, 0.75]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0.25 , 0.375],
[0.75 , 0.375]])

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [1.5, 3.0],
...     [3.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [ 3.0  ,  1.5    ],
...     [ 2.625, -0.90625],
...     [-0.75 ,  2.4375 ],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.25 , 0.75 ],
[0.875, 0.25 ]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0.75  , 1.125  ],
[2.625 , 0.65625]])

>>> nodes1 = np.asfortranarray([
...     [0.0  , 0.0  ],
...     [0.375, 0.75 ],
...     [0.75 , 0.375],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.25 , 0.5625],
...     [0.625, 0.1875],
...     [1.0  , 0.9375],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.5       , 0.16666...],
[0.83333..., 0.5       ]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0.375 , 0.46875],
[0.625 , 0.46875]])


Even for curves which don’t intersect at exact floating point numbers, we can compute the intersection to machine precision:

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [1.125,  0.5],
...     [0.625, -0.5],
...     [0.125,  0.5],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> sq31 = np.sqrt(31.0)
>>> expected_ints = np.asfortranarray([
...     [9 - sq31, 9 + sq31],
...     [9 + sq31, 9 - sq31],
... ]) / 16.0
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err) <= -53
True
>>> points = curve1.evaluate_multi(intersections[:, 0])
>>> expected_pts = np.asfortranarray([
...     [36 - 4 * sq31, 16 + sq31],
...     [36 + 4 * sq31, 16 - sq31],
... ]) / 64.0
>>> max_err = np.max(np.abs(points - expected_pts))
>>> binary_exponent(max_err)
-54

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.0, 0.265625],
...     [0.5, 0.234375],
...     [1.0, 0.265625],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> sq33 = np.sqrt(33.0)
>>> expected_ints = np.asfortranarray([
...     [33 - 4 * sq33, 33 - 4 * sq33],
...     [33 + 4 * sq33, 33 + 4 * sq33],
... ]) / 66.0
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err)
-55
>>> points = curve1.evaluate_multi(intersections[:, 0])
>>> expected_pts = np.asfortranarray([
...     [33 - 4 * sq33, 17],
...     [33 + 4 * sq33, 17],
... ]) / 66.0
>>> max_err = np.max(np.abs(points - expected_pts))
>>> binary_exponent(max_err)
-54

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.0 ,  0.0],
...     [0.25,  2.0],
...     [0.5 , -2.0],
...     [0.75,  2.0],
...     [1.0 ,  0.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=4)
>>> intersections = curve1.intersect(curve2)
>>> points = curve1.evaluate_multi(intersections[:, 0])
>>> sq7 = np.sqrt(7.0)
>>> expected_ints = np.asfortranarray([
...     [7 - sq7, 7 - sq7],
...     [7 + sq7, 7 + sq7],
...     [      0, 0      ],
...     [     14, 14     ],
... ]) / 14.0
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err) <= -53
True
>>> expected_pts = np.asfortranarray([
...     [7 - sq7, 6],
...     [7 + sq7, 6],
...     [      0, 0],
...     [     14, 0],
... ]) / 14.0
>>> max_err = np.max(np.abs(points - expected_pts))
>>> binary_exponent(max_err) <= -53
True

>>> nodes1 = np.asfortranarray([
...     [-0.125, -0.28125],
...     [ 0.5  ,  1.28125],
...     [ 1.125, -0.28125],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [ 1.5625, -0.0625],
...     [-1.5625,  0.25  ],
...     [ 1.5625,  0.5625],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> sq5 = np.sqrt(5.0)
>>> expected_ints = np.asfortranarray([
...     [4 - sq5, 6 - sq5],
...     [      3, 7      ],
...     [      9, 1      ],
...     [4 + sq5, 6 + sq5],
... ]) / 10.0
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err) <= -51
True
>>> points = curve1.evaluate_multi(intersections[:, 0])
>>> expected_pts = np.asfortranarray([
...     [6 - 2 * sq5, 5 - sq5],
...     [          4, 6      ],
...     [         16, 0      ],
...     [6 + 2 * sq5, 5 + sq5],
... ]) / 16.0
>>> max_err = np.max(np.abs(points - expected_pts))
>>> binary_exponent(max_err)
-51


For higher degree intersections, the error starts to get a little larger.

>>> nodes1 = np.asfortranarray([
...     [0.25 , 0.625],
...     [0.625, 0.25 ],
...     [1.0  , 1.0  ],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.0 , 0.5],
...     [0.25, 1.0],
...     [0.75, 1.5],
...     [1.0 , 0.5],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=3)
>>> intersections = curve1.intersect(curve2)
>>> s_vals = np.roots([486, -3726, 13905, -18405, 6213, 1231])
>>> _, s_val, _ = np.sort(s_vals[s_vals.imag == 0].real)
>>> t_vals = np.roots([4, -16, 13, 25, -28, 4])
>>> _, _, t_val = np.sort(t_vals[t_vals.imag == 0].real)
>>> expected_ints = np.asfortranarray([
...     [s_val, t_val],
... ])
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err)
-50
>>> points = curve1.evaluate_multi(intersections[:, 0])
>>> x_val = (3 * s_val + 1) / 4
>>> y_val = (9 * s_val * s_val - 6 * s_val + 5) / 8
>>> expected_pts = np.asfortranarray([
...     [x_val, y_val],
... ])
>>> max_err = np.max(np.abs(points - expected_pts))
>>> binary_exponent(max_err) <= -50
True

>>> nodes1 = np.asfortranarray([
...     [0.0, 8.0],
...     [6.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=1)
>>> nodes2 = np.asfortranarray([
...     [0.375, 7.0],
...     [2.125, 8.0],
...     [3.875, 0.0],
...     [5.625, 1.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=3)
>>> intersections = curve1.intersect(curve2)
>>> sq7 = np.sqrt(7.0)
>>> expected_ints = np.asfortranarray([
...     [          24, 24          ],
...     [24 - 7 * sq7, 24 - 8 * sq7],
...     [24 + 7 * sq7, 24 + 8 * sq7],
... ]) / 48.0
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err)
-53
>>> points = curve1.evaluate_multi(intersections[:, 0])
>>> expected_pts = np.asfortranarray([
...     [           72, 96           ],
...     [72 - 21 * sq7, 96 + 28 * sq7],
...     [72 + 21 * sq7, 96 - 28 * sq7],
... ]) / 24.0
>>> max_err = np.max(np.abs(points - expected_pts))
>>> binary_exponent(max_err)
-50

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.375],
...     [1.0, 0.375],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=1)
>>> nodes2 = np.asfortranarray([
...     [0.125, 0.25  ],
...     [0.375, 0.75  ],
...     [0.625, 0.0   ],
...     [0.875, 0.1875],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=3)
>>> intersections = curve1.intersect(curve2)
>>> points = curve1.evaluate_multi(intersections[:, 0])
>>> s_val2, s_val1, _ = np.sort(np.roots(
...     [17920, -29760, 13512, -1691]))
>>> t_val2, t_val1, _ = np.sort(np.roots([35, -60, 24, -2]))
>>> expected_ints = np.asfortranarray([
...     [s_val1, t_val1],
...     [s_val2, t_val2],
... ])
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err)
-51
>>> expected_pts = np.asfortranarray([
...     [s_val1, 0.375],
...     [s_val2, 0.375],
... ])
>>> max_err = np.max(np.abs(points - expected_pts))
>>> binary_exponent(max_err)
-51


## Intersections at Endpoints¶

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [1.0,  0.0],
...     [1.5, -1.0],
...     [2.0,  0.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[1., 0.]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[1., 0.]])

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [2.0, 0.0],
...     [1.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[1., 1.]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[1., 0.]])

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [4.5, 9.0],
...     [9.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [11.0,  8.0],
...     [ 7.0, 10.0],
...     [ 3.0,  4.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.3333..., 1. ]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[3., 4.]])


## Detecting Self-Intersections¶

>>> nodes = np.asfortranarray([
...     [ 0.0 , 2.0  ],
...     [-1.0 , 0.0  ],
...     [ 1.0 , 1.0  ],
...     [-0.75, 1.625],
... ])
>>> curve = bezier.Curve(nodes, degree=3)
>>> left, right = curve.subdivide()
>>> intersections = left.intersect(right)
>>> sq5 = np.sqrt(5.0)
>>> expected_ints = np.asfortranarray([
...     [      3, 0  ],
...     [3 - sq5, sq5],
... ]) / 3.0
>>> max_err = np.max(np.abs(intersections - expected_ints))
>>> binary_exponent(max_err)
-54
>>> left.evaluate_multi(intersections[:, 0])
array([[-0.09375 , 0.828125],
[-0.25    , 1.375   ]])


## Limitations¶

Intersections that occur at points of tangency are in general problematic. For example, consider

$\begin{split}B_1(s) = \left[ \begin{array}{c} s \\ 2s(1 - s)\end{array}\right], \quad B_2(t) = \left[ \begin{array}{c} t \\ t^2 + (1 - t)^2 \end{array}\right]\end{split}$

The first curve is the zero set of $$y - 2x(1 - x)$$, so plugging in the second curve gives

$0 = t^2 + (1 - t)^2 - 2t(1 - t) = (2t - 1)^2.$

This shows that a point of tangency is equivalent to a repeated root of a polynomial. For this example, the intersection process successfully terminates

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.0, 1.0],
...     [0.5, 0.0],
...     [1.0, 1.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.5, 0.5]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[0.5, 0.5]])


However this library mostly avoids (for now) computing tangent intersections. For example, the curves

have a tangent intersection that this library fails to compute:

>>> nodes1 = np.asfortranarray([
...     [0.0  , 0.0  ],
...     [0.375, 0.75 ],
...     [0.75 , 0.375],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.25 , 0.625],
...     [0.625, 0.25 ],
...     [1.0  , 1.0  ],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> curve1.intersect(curve2)
Traceback (most recent call last):
...
NotImplementedError: Line segments parallel.


This failure comes from the fact that the linear approximations of the curves near the point of intersection are parallel.

As above, we can find some cases where tangent intersections are resolved:

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [4.5, 9.0],
...     [9.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [3.0, 4.5],
...     [8.0, 4.5],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=1)
>>> intersections = curve1.intersect(curve2)
>>> intersections
array([[0.5, 0.3]])
>>> curve1.evaluate_multi(intersections[:, 0])
array([[4.5, 4.5]])


but even by rotating an intersection (from above) that we know works

we still see a failure

>>> nodes1 = np.asfortranarray([
...     [ 0.0, 0.0],
...     [-0.5, 1.5],
...     [ 1.0, 1.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [-1.0, 1.0],
...     [ 0.5, 0.5],
...     [ 0.0, 2.0],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> curve1.intersect(curve2)
Traceback (most recent call last):
...
NotImplementedError: The number of candidate intersections is too high.


In addition to points of tangency, coincident curve segments are (for now) not supported. For the curves

the library fails as well

>>> nodes1 = np.asfortranarray([
...     [0.0, 0.0],
...     [0.5, 1.0],
...     [1.0, 0.0],
... ])
>>> curve1 = bezier.Curve(nodes1, degree=2)
>>> nodes2 = np.asfortranarray([
...     [0.25,  0.375],
...     [0.75,  0.875],
...     [1.25, -0.625],
... ])
>>> curve2 = bezier.Curve(nodes2, degree=2)
>>> curve1.intersect(curve2)
Traceback (most recent call last):
...
NotImplementedError: The number of candidate intersections is too high.